Multiplying Rational Expressions
April 26, 2025
INTRODUCTION
Back in elementary, we learned that multiplying fractions is simple: multiply the numerators together, then the denominators. After that, we simplify the result by canceling out any common factors. The example below shows how this works step by step.
\textbf{Simplify} \displaystyle \dfrac{3}{5} \cdot \dfrac{5}{9}.
\textbf{Solution:} \displaystyle \frac{3}{5} \cdot \frac{5}{9} = \frac{3 \cdot 5}{5 \cdot 9} = \frac{\cancel{3}^1 \cdot \cancel{5}^1}{\cancel{5}^1 \cdot \cancel{3}^1 \cdot 3} = \boxed{\frac{1}{3}}
Multiplying rational expressions works just like multiplying fractions. First, factor all terms in the numerators and denominators completely. Then, cancel out any common factors that appear in both the numerator and denominator. Finally, multiply the remaining factors across the top and bottom.
Suppose P, Q, R, \text{ and } S are polynomials. When multiplying two rational expressions, multiply the numerators together and multiply the denominators together.
where Q \neq 0 \text { and } S \neq 0.
For simplicity, multiplying rational expressions are done using these simple steps:
Step 1: Factor all numerators and denominators completely.
Step 2: Cancel any common factors between numerators and denominators.
Step 3: Multiply the remaining factors across the numerator and denominator.
